arXiv subjects
Publications and source records attributed to Henrique Gomes.
We provide a unified geometrical origin for both boundary charges and particle dressings, with a focus on electrodynamics. The method is furthermore generalizable to QCD and gravity, and can be extended to the non-perturbative domain.
On the path towards quantum gravity, we find friction between temporal relations in quantum mechanics (QM) (where they are fixed and field-independent), and in general relativity (where they are field-dependent and dynamic). This paper aims to attenuate that friction, by encoding gravity in the timeless configuration space of spatial fields with dynamics given by a path integral. The framework demands that boundary conditions for this path integral be uniquely given, but unlike other approaches where they are prescribed --- such as the no-boundary and the tunneling proposals --- here I postulate basic principles to identify boundary conditions in a large class of theories. Uniqueness arises only if a reduced configuration space can be defined and if it has a profoundly asymmetric fundamental structure. These requirements place strong restrictions on the field and symmetry content of theories encompassed here; shape dynamics is one such theory. Also as in other boundary proposals, Time, including space-time, emerges as an effective concept; valid for certain curves in configuration space but not assumed from the start. When some such notion of time becomes available, conservation of (positive) probability currents ensues. I show that, in the appropriate limits, a Schroedinger equation dictates the evolution of weakly coupled source fields on a classical gravitational background. Due to the asymmetry of reduced configuration space, these probabilities and currents avoid a known difficulty of standard WKB approximations for Wheeler DeWitt in minisuperspace: the selection of a unique Hamilton-Jacobi solution to serve as background. I illustrate these constructions with a simple example of a full quantum gravitational theory (i.e. not in minisuperspace) for which the formalism is applicable, and give a formula for calculating gravitational semi-classical relative probabilities in it.
The cuscuton was introduced in the context of cosmology as a field with infinite speed of propagation. It has been claimed to resemble Hořava gravity in a certain limit, and it is a good candidate for an ether theory in which a time-dependent cosmological constant appears naturally. The analysis of its properties is usually performed in the Lagrangian framework, which makes issues like the counting of its dynamical degrees of freedom less clear-cut. Here we perform a Hamiltonian analysis of the theory. We show that the homogeneous limit with local degrees of freedom has singular behavior in the Hamiltonian framework. In other frames, it has an extra scalar degree of freedom. The homogeneous field has regular behavior only if defined a priori as a spatially constant field in a CMC foliation and contributing with a single global degree of freedom. Lastly, we find conditions on the cuscuton potential for the resulting lapse function to be non-zero throughout evolution.
We introduce a functional covariant differential as a tool for studying field space geometry in a manifestly covariant way. We then touch upon its role in gauge theories and general relativity over bounded regions, and in BRST symmetry. Due to the Gribov problem, we argue that our formalism ---allowing for a non-vanishing functional curvature---is necessary for a global treatment of gauge-invariance in field space. We conclude by suggesting that the structures we introduce satisfactorily implement the notion of a (non-asymptotic) observer in gauge theories and general relativity.
We give the most general conditions to date which lead to uniqueness of the general relativistic Hamiltonian. Namely, we show that all spatially covariant generalizations of the scalar constraint which extend the standard one while remaining quadratic in the momenta are second class. Unlike previous investigations along these lines, we do not require a specific Poisson bracket algebra, and the quadratic dependence on the momenta is completely general, with an arbitrary local operator as the kinetic term.
In a recent paper, one of us studied spherically symmetric, asymptotically flat solutions of Shape Dynamics, finding that the spatial metric has characteristics of a wormhole - two asymptotically flat ends and a minimal-area sphere, or `throat', in between. In this paper we investigate whether that solution can emerge as a result of gravitational collapse of matter. With this goal, we study the simplest kind of spherically-symmetric matter: an infinitely-thin shell of dust. Our system can be understood as a model of a star accreting a thin layer of matter. We solve the dynamics of the shell exactly and find that, indeed, as it collapses, the shell leaves in its wake the wormhole metric. In the maximal-slicing time we use for asymptotically flat solutions, the shell only approaches the throat asymptotically and does not cross it in a finite amount of time (as measured by a clock `at infinity'). This leaves open the possibility that a more realistic cosmological solution of Shape Dynamics might see this crossing happening in a finite amount of time (as measured by the change of relational/shape degrees of freedom).
In a previous paper, we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe $M$, existing on timeless configuration space $\mathcal{M}$. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. To deal with the challenges gauge symmetries may pose to a good definition of locality, I start by demanding symmetries to have an action on $\mathcal{M}$ so that the quotient wrt the symmetries respects certain factorizations of $\mathcal{M}$. These factorizations are algebraic splits of $\mathcal{M}$ into sub-spaces $\mathcal{M}=\bigoplus_i \mathcal{M}_{O_i}$-- each factor corresponding to a physical sub-region $O_i$. This deals with kinematic locality, but locality in full can only emerge dynamically, and is not postulated. I describe conditions under which it can be said to have emerged. The dynamics of $O$ is independent of its complement, $M-O$, if the projection of extremal curves on $M$ onto the space of extremal curves intrinsic to $O$ is a surjective map. This roughly corresponds to $e^{i\hat{H}t}\circ \mathsf{pr}_{\mbox{O}}= \mathsf{pr}_{\mbox{ O}}\circ e^{i\hat{H}t}$, where $\mathsf{pr}_{\mbox{ O}}:\mathcal{M}\rightarrow \mathcal{M}_O^{\partial O}$ is a linear projection. This criterion for locality can be made approximate -- an impossible feat had it been already postulated -- and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are \emph{mutually independent} according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is what one would like to obtain from any definition of locality.
We describe a class of modified gravity theories that deform general relativity in a way that breaks time reversal invariance and, very mildly, locality. The algebra of constraints, local physical degrees of freedom, and their linearized equations of motion, are unchanged, yet observable effects may be present on cosmological scales, which have implications for the early history of the universe. This is achieved in the Hamiltonian framework, in a way that requires the constant mean curvature gauge conditions and is, hence, inspired by shape dynamics.
We provide a bottom-up argument to derive some known results from holographic renormalization using the classical bulk-bulk equivalence of General Relativity and Shape Dynamics, a theory with spatial conformal (Weyl) invariance. The purpose of this paper is twofold: 1) to advertise the simple classical mechanism: trading of gauge symmetries, that underlies the bulk-bulk equivalence of General Relativity and Shape Dynamics to readers interested in dualities of the type of AdS/CFT; and 2) to highlight that this mechanism can be used to explain certain results of holographic renormalization, providing an alternative to the AdS/CFT conjecture for these cases. To make contact with usual the semiclassical AdS/CFT correspondence, we provide, in addition, a heuristic argument that makes it plausible why the classical equivalence between General Relativity and Shape Dynamics turns into a duality between radial evolution in gravity and the renormalization group flow of a conformal field theory. We believe that Shape Dynamics provides a new perspective on gravity by giving conformal structure a primary role within the theory. It is hoped that this work provides the first steps towards understanding what this new perspective may be able to teach us about holographic dualities.
There exist several ways of constructing general relativity from `first principles': Einstein's original derivation, Lovelock's results concerning the exceptional nature of the Einstein tensor from a mathematical perspective, and Hojman-Kuchař-Teitelboim's derivation of the Hamiltonian form of the theory from the symmetries of spacetime, to name a few. Here I propose a different set of first principles to obtain general relativity in the canonical framework without presupposing spacetime in any way. I first require consistent propagation of scalar spatially covariant constraints. I find that up to a certain order in derivatives (four spatial and two temporal), there are large families of such consistently propagated constraints. Then I look for pairs of such constraints that can gauge-fix each other and form a theory with two dynamical degrees of freedom per \emph{space} point. This demand singles out the ADM Hamiltonian either in i) CMC gauge, with arbitrary (finite, non-zero) speed of light, and an extra term linear in York time, or ii) a gauge where the Hubble parameter is conformally harmonic.
Shape dynamics is a classical theory of gravity which agrees with general relativity in many important aspects, but which possesses different gauge symmetries and can present some fundamental global differences with respect to Einstein spacetimes. Here, we present a general procedure for (locally) mapping stationary, axisymmetric general relativity solutions onto their shape dynamic counterparts. We focus in particular on the rotating black hole solution for shape dynamics and show that many of the properties of the spherically symmetric solution are preserved in the extension to the axisymmetric case: it is also free of physical singularities, it does not form a space-time at the horizon, and it possesses an inversion symmetry about the horizon which leads to us to interpret the solution as a wormhole.
Shape Dynamics is a theory of gravity that replaces refoliation invariance for spatial Weyl invariance. Those solutions of the Einstein equations that have global, constant mean curvature slicings, are mirrored by solutions in Shape Dynamics. However, there are solutions of Shape Dynamics that have no counterpart in General relativity, just as there are solutions of GR that are not completely foliable by global constant mean curvature slicings (such as the Schwarzschild spacetime). It is therefore interesting to analyze directly the equations of motion of Shape Dynamics in order to find its own solutions, irrespective of properties of known solutions of GR. Here I perform a first study in this direction by utilizing the equations of motion of Shape Dynamics in a spherically symmetric, asymptotically flat ansatz to derive an analogue of the Birkhoff theorem. There are two significant differences with respect to the usual Birkhoff theorem in GR. The first regards the construction of the solution: the spatial Weyl gauge freedom of shape dynamics is used to simplify the problem, and boundary conditions are required. In fact the derivation is simpler than the usual Birkhof theorem as no Christoffel symbols are needed. The second, and most important difference is that the solution obtained is uniquely the isotropic wormhole solution, in which no singularity is present, as opposed to maximally extended Schwarzschild. This provides an explicit example of the breaking of the duality between General relativity and Shape Dynamics, and exhibits some of its consequences.
The differing concepts of time in general relativity and quantum mechanics are widely accused as the main culprits in our persistent failure in finding a complete theory of quantum gravity. Here we address this issue by constructing ADM-type theories \emph{in a particular time gauge} directly from first principles. The principles are expressed as conditions on phase space constraints: we search for two sets of spatially covariant constraints, which generate symmetries (are first class) and gauge-fix each other leaving two propagating degrees of freedom. One of the sets is the Weyl generator tr$(π)$, and the other is a one-parameter family containing the ADM scalar constraint $λR- β(π^{ab}π_{ab}+(\mbox{tr}(π))^2/2))$. The two sets of constraints can be seen as defining ADM-type theories with a maximal slicing gauge-fixing. This work provides an independent, first principles derivation of ADM gravity. The principles above are motivated by a heuristic argument relying in the relation between symmetry doubling and exact renormalization arguments for quantum gravity, aside from compatibility with the spatial diffeomorphisms. As a by-product, these results address one of the most popular criticisms of Shape Dynamics: its construction starts off from the ADM Hamiltonian formulation. The present work severs this dependence: the set of constraints yield reduced phase space theories that can be naturally represented by either Shape Dynamics or ADM. More precisely, the resulting theories can be naturally "unfixed" to encompass either spatial Weyl invariance (the symmetry of Shape Dynamics) or refoliation symmetry (ADM).
One of the more popular objections towards shape dynamics is the suspicion that anomalies in the spatial Weyl symmetry will arise upon quantization. The purpose of this short paper is to establish the tools required for an investigation of the sort of anomalies that can possibly arise. The first step is to adapt to our setting Barnich and Henneaux's formulation of gauge cohomology in the Hamiltonian setting, which serve to decompose the anomaly into a spatial component and time component. The spatial part of the anomaly, i.e. the anomaly in the symmetry algebra itself ($[Ω, Ω]\propto \hbar$ instead of vanishing) is given by a projection of the second ghost cohomology of the Hamiltonian BRST differential associated to $Ω$, modulo spatial derivatives. The temporal part, $[Ω, H]\propto\hbar$ is given by a different projection of the first ghost cohomology and an extra piece arising from a solution to a functional differential equation. Assuming locality of the gauge cohomology groups involved, this part is always local. Assuming locality for the gauge cohomology groups, using Barnich and Henneaux's results, the classification of Weyl cohomology for higher ghost numbers performed by Boulanger, and following the descent equations, we find a complete characterizations of anomalies in 3+1 dimensions. The spatial part of the anomaly and the first component of the temporal anomaly are always local given these assumptions even in shape dynamics. The part emerging from the solution of the functional differential equations explicitly involves the shape dynamics Hamiltonian, and thus might be non-local. If one restricts this extra piece of the temporal anomaly to be also local, then overall no \emph{local} Weyl anomalies, either temporal or spatial, emerge in the 3+1 case.
Shape Dynamics is a theory of gravity that waives refoliation invariance in favor of spatial Weyl invariance. It is a canonical theory, constructed from a Hamiltonian, 3+1 perspective. One of the main deficits of Shape Dynamics is that its Hamiltonian is only implicitly constructed as a functional of the phase space variables. In this paper, I write down the equations of motion for Shape Dynamics to show that over a curve in phase space representing a Minkowski spacetime, Shape Dynamics possesses Poincaré symmetry for appropriate boundary conditions. The proper treatment of such boundary conditions leads us to completely formulate Shape Dynamics for open manifolds in the asymptotically flat case. We study the charges arising in this case and find a new definition of total energy, which is completely invariant under spatial Weyl transformations close to the boundary. We then use the equations of motion once again to find a non-trivial solution of Shape Dynamics, consisting of a flat static Universe with a point-like mass at the center. We calculate its energy through the new formula and rederive the usual Schwarzschild mass.
The purpose of this contribution is to elucidate some of the properties of Shape Dynamics (SD) and is largely based on a recent longer article. We shall point out some of the key differences between SD and related theoretical constructions, illustrate the central mechanism of symmetry trading in electromagnetism and finally point out some new quantization strategies inspired by SD. We refrain from describing mathematical detail and from citing literature. For both we refer to the longer article.
The dynamics of gravity can be described by two different systems. The first is the familiar spacetime picture of General Relativity, the other is the conformal picture of Shape Dynamics. We argue that the bulk equivalence of General Relativity and Shape Dynamics is a natural setting to discuss familiar bulk/boundary dualities. We discuss consequences of the Shape Dynamics description of gravity as well as the issue why the bulk equivalence is not explicitly seen in the General Relativity description of gravity.
General Relativity can be formulated in terms of a spatially Weyl invariant gauge theory called Shape Dynamics. Using this formulation, we establish a "bulk/bulk" duality between gravity and a Weyl invariant theory on spacelike Cauchy hypersurfaces. This duality has two immediate consequences: i) it leads trivially to a corresponding "bulk/boundary" duality between General Relativity and a boundary CFT, and ii) the boundary can be defined in a gauge-invariant way. Moreover, the corresponding bulk/boundary duality is sufficient to explain a large portion of the evidence in favor of gauge/gravity duality and provides independent evidence for the AdS/CFT correspondence.